What should not be aspired to? Travelling to heavens/buddha-fields, talking to highly realised beings, or clarifying doubts? It is said that it worked for Asanga.

1Myriad dharmas are only mind.
Mind is unobtainable.
What is there to seek?2If the Buddha-Nature is seen,
there will be no seeing of a nature in any thing.3Neither cultivation nor seated meditation —
this is the pure Chan of Tathagata.4With sudden enlightenment to Tathagata Chan,
the six paramitas and myriad means
are complete within that essence.

What should not be aspired to? Travelling to heavens/buddha-fields, talking to highly realised beings, or clarifying doubts? It is said that it worked for Asanga.

Well, what you would you suggest to young Matthew here. How could he visit 'Sukhavati heaven' in a reasonable time-frame - like, a few months, or a year maybe? Any tips?

No wisdom can we get hold of
No highest perfection
No Bodhisattva, no thought of enlightenment either
When told of this, if not bewildered and in no way anxious
A Bodhisattva courses in the Tathāgata's wisdom.

1Myriad dharmas are only mind.
Mind is unobtainable.
What is there to seek?2If the Buddha-Nature is seen,
there will be no seeing of a nature in any thing.3Neither cultivation nor seated meditation —
this is the pure Chan of Tathagata.4With sudden enlightenment to Tathagata Chan,
the six paramitas and myriad means
are complete within that essence.

No wisdom can we get hold of
No highest perfection
No Bodhisattva, no thought of enlightenment either
When told of this, if not bewildered and in no way anxious
A Bodhisattva courses in the Tathāgata's wisdom.

How can one stay awake for three months while constantly moving ?
Is that a metaphor for something?

“The path of the supreme yoga it is not the path of accomplished sages of the past. Whoever enters onto the path of the sages of the past will end up gripped by the sicknesses of the path - meditation, attachment, and exertion.”Thig le drug pa.

“Everything of the universe of saṃsāra and nirvāṇa arises as the enlightened energy of the one self-perfected Natural Presence. But these teachers still mistakenly teach that disciples should fabricate enlightenment by applying discipline, renunciation, interruption, purification and transformation”.

How can one stay awake for three months while constantly moving ?
Is that a metaphor for something?

The sutra says 7 days.
The Kaihogyo that Tendai monks do now is about 1000 days, but only about 9 days no sleep.
The Pratyutpanna-based practice devised by Zhiyi is 90, but I believe there are designated sleep periods during each of those 90 days and that it is only during practice periods that one may not relax.

How can one stay awake for three months while constantly moving ?
Is that a metaphor for something?

The sutra says 7 days.
The Kaihogyo that Tendai monks do now is about 1000 days, but only about 9 days no sleep.
The Pratyutpanna-based practice devised by Zhiyi is 90, but I believe there are designated sleep periods during each of those 90 days and that it is only during practice periods that one may not relax.

Ok but this sutra linked by Astus, in the chapter on the four things says the following:

Chapter III
The Four Things
[The Buddha said:]
If bodhisattvas possess four things, or dharmas, they quickly master
this meditation. What are the four? First, no one can destroy their faith.
Second, no one can withstand their energy. Third, no one can match
the wisdom they attain. Fourth, they always devote themselves to good teachers.
These are the four.

If bodhisattvas possess a further four things they quickly master this meditation.
What are the four? First, for three months they should
not have a worldly thought, even for the time it takes to snap the fingers.
Second, for three months they should not go to sleep, even for the time
it takes to snap the fingers. Third, continually walking, they should not
stop or sit down for three months, except for around mealtimes. Fourth,
in preaching the sutras to others, they should not expect clothes or food
and drink from them. These are the four.

“The path of the supreme yoga it is not the path of accomplished sages of the past. Whoever enters onto the path of the sages of the past will end up gripped by the sicknesses of the path - meditation, attachment, and exertion.”Thig le drug pa.

“Everything of the universe of saṃsāra and nirvāṇa arises as the enlightened energy of the one self-perfected Natural Presence. But these teachers still mistakenly teach that disciples should fabricate enlightenment by applying discipline, renunciation, interruption, purification and transformation”.

According to the teachings heard, one should also think of His land called Sukhāvatī, which is ten million koṭi Buddha Lands away from here. One should single-mindedly contemplate for one day and one night, or even seven days and seven nights. After the seventh day, one will see Him.

The Chinese version says:
"There are another four things which will enable one to attain this samādhi quickly. First, do not engage in worldly thinking for three months, not even during a finger snap. Second, do not sleep for three months, not even during a finger snap. Third, do walking meditation for three months without any rest, except when eating and so forth. Fourth, expound sūtras to others, not expecting their offerings. These are the four things."

"and so forth" could mean allotted hours for sleep/rest at night.

In practice, it seems more likely sleep was done, albeit in a seated meditation posture.

We read that the Buddha would sometimes take an afternoon nap or siesta (divàseyyà), although this was probably only after he had become quite old. An ascetic named Saccaka once asked the Buddha if he slept in the afternoon and he replied: `I can recall that in the last month of the hot season after returning from my alms round and having eaten my meal that I would fold my robe into four, spread it out, lie down and go to sleep mindfully and fully aware.' Saccaka sniffed: `Some would call that abiding in delusion' (M.I,250).

The Buddha observed that our behaviour while awake can sometimes have an effect during sleep. Being immoral or cruel, for example, can cause insomnia or restless sleep. `When the fool is on his couch or bed or lying on the ground, the evil actions he has done in the past descend on him, settle on him, lie on him, just as the shadow of a mountain in the late afternoon descends, settles and lies on the earth. At that time the fool thinks ßI have not done what is beautiful or skilful. I have not protected myself against the fearful û Then he is uneasy and troubled, he weeps and despairs' (condensed, M.III,164-5). Likewise, a person who has a loving disposition `sleeps happily, wakes happily and has no bad dreams' (sukhaü supati, sukhaü pañibhujjhati, na pàpakaü supinaü passati, A.V.342).

The Buddha said that sound and peaceful rest is one of the many positive results of being enlightened. `The enlightened one, done with sensual pleasures, free from acquisitions and with a cool heart, always sleeps happily. When attachments are chopped off, the heart is carefree, and the mind is at peace, and the serene one sleeps well'(S.I,212).

I don't believe the Pratyutpanna Samadhi teaches how to travel to other pure realms; rather, that samadhi seems specific to receiving visions of Buddhas, without coming or going anywhere (and with the Buddhas neither coming nor going anywhere). But obviously strengthening your connection to Amitabha can only help.

The only sure-fire method I know of to enter the heavens is through Thien/zazen, and acquiring the siddhi of the mind-made body (which, in the EBTs, I think roughly correlates with mastering the four dhyanas). Not to say that others don't exist, but this is the only method that I have personally been made aware of in my education.

But really, Pure Land experiences -- dropping off the heavenly experiences, since those seem quite a bit rarer and more difficult to achieve -- seems to simply come as the result of diligent cultivation. Do the practices and really try to remain mindful of the Buddha at all times (beyond just chanting his name, really try to connect with the spirit of what 'Buddha-mindfulness' means). Some individuals may never have Pure Land experiences in their lifetime, and some may have experiences quickly, but the practice is the key either way.

The method of mastering the jhanas would help you to reach heavenly realms within samsara, not further than that.

The method of mastering the jhanas would help you to reach heavenly realms within samsara, not further than that.

Maybe according to some mahayana source.
But if you look at the buddha's own talking about his enlightenment or into the pali canon, it is very clear that jhana is of paramount importance.

"I thought: 'I recall once, when my father the Sakyan was working, and I was sitting in the cool shade of a rose-apple tree, then — quite secluded from sensuality, secluded from unskillful mental qualities — I entered & remained in the first jhana: rapture & pleasure born from seclusion, accompanied by directed thought & evaluation. Could that be the path to Awakening?' Then following on that memory came the realization: 'That is the path to Awakening.'

The method of mastering the jhanas would help you to reach heavenly realms within samsara, not further than that.

Maybe according to some mahayana source.
But if you look at the buddha's own talking about his enlightenment or into the pali canon, it is very clear that jhana is of paramount importance.

"I thought: 'I recall once, when my father the Sakyan was working, and I was sitting in the cool shade of a rose-apple tree, then — quite secluded from sensuality, secluded from unskillful mental qualities — I entered & remained in the first jhana: rapture & pleasure born from seclusion, accompanied by directed thought & evaluation. Could that be the path to Awakening?' Then following on that memory came the realization: 'That is the path to Awakening.'

I don't believe the Pratyutpanna Samadhi teaches how to travel to other pure realms; rather, that samadhi seems specific to receiving visions of Buddhas, without coming or going anywhere (and with the Buddhas neither coming nor going anywhere). But obviously strengthening your connection to Amitabha can only help.

The only sure-fire method I know of to enter the heavens is through Thien/zazen, and acquiring the siddhi of the mind-made body (which, in the EBTs, I think roughly correlates with mastering the four dhyanas). Not to say that others don't exist, but this is the only method that I have personally been made aware of in my education.

But really, Pure Land experiences -- dropping off the heavenly experiences, since those seem quite a bit rarer and more difficult to achieve -- seems to simply come as the result of diligent cultivation. Do the practices and really try to remain mindful of the Buddha at all times (beyond just chanting his name, really try to connect with the spirit of what 'Buddha-mindfulness' means). Some individuals may never have Pure Land experiences in their lifetime, and some may have experiences quickly, but the practice is the key either way.

The method of mastering the jhanas would help you to reach heavenly realms within samsara, not further than that.

I read OP's query as asking about heavenly realms AND Pure Lands, with those being different concepts. I don't consider Pure Lands to be heavenly realms (hence "dropping off heavenly realms" when I reference that in the above post), so I was answering one-half of what I perceived the question to be, and then answering the other half separately.